ABSTRACT
Systems whose arrival or service rates fluctuate over time are very common, but are still not well understood analytically. Stationary formulas are poor predictors of systems with fluctuating load. When the arrival and service processes fluctuate in a Markovian manner, computational methods, such as Matrix-analytic and spectral analysis, have been instrumental in the numerical evaluation of quantities like mean response time. However, such computational tools provide only limited insight into the functional behavior of the system with respect to its primitive input parameters: the arrival rates, service rates, and rate of fluctuation.For example, the shape of the function that maps rate of fluctuation to mean response time is not well understood, even for an M/M/1 system. Is this function increasing, decreasing, monotonic? How is its shape affected by the primitive input parameters? Is there a simple closed-form approximation for the shape of this curve? Turning to user experience: How is the performance experienced by a user arriving into a "high load" period different from that of a user arriving into a "low load" period, or simply a random user. Are there stochastic relations between these? In this paper, we provide the first answers to these fundamental questions.
- J. Abate, G. Choudhury, W. Whitt, Asymptotics for Steady-State Tail Probabilities in Structured Markov Queueing Models, Commun. Statist.-Stoch. Mod. 10(1), pp. 99--143, 1994.Google Scholar
- I. J. B. F. Adan, V. G. Kulkarni. Single-Server Queue with Markov-Dependent Inter-Arrival and Service Times,QUESTA 45, pp. 113--134, 2003. Google ScholarDigital Library
- E. Arjas. On the Use of a Fundamental Identity in the Theory of Semi-Markov Queues, Adv. Appl. Prob. 4, pp. 271--284, 1972.Google ScholarCross Ref
- S. Asmussen, J. Møller. Calculation of the Steady State Waiting Time Distribution in GI/PH/c and MAP/PH/c Queues, QUESTA 37, pp. 9--29, 2001. Google ScholarDigital Library
- N. T. J. Bailey. A continuous time treatment of a simple queue using generating functions, J. R. Statist. Soc. B19, pp. 326--333, 1954.Google Scholar
- G. L. Choudhury, A. Mandelbaum, M. I. Reiman, W. Whitt. Fluid and Diffusion Limits for Queues in Slowly Changing Environments, Stoch. Mod. 13, pp. 121--146, 1997.Google ScholarCross Ref
- E. Çinlar. Time Dependence of Queues with Semi-Markov Services, J. Appl. Prob. 4, pp. 356--364, 1967.Google ScholarCross Ref
- E. Çinlar. Queues with Semi-Markov Arrivals, J. Appl. Prob. 4, pp. 365--379, 1967.Google ScholarCross Ref
- A. B. Clark. A Waiting Line Process of the Markov Type, Ann. Math. Statist. 27, pp. 452--459, 1956.Google ScholarCross Ref
- J. H. A. de Smit. The Single Server Semi-Markov Queue, Stoch. Proc. and Appl. 22, pp. 37--50, 1986.Google ScholarCross Ref
- E. Gelenbe, C. Rosenberg. Queues with Slowly Varying Arrival and Service Processes, Man. Sci. 36(8), pp. 928--937, 1990. Google ScholarDigital Library
- V. Gupta, M. Harchol-Balter, A. Scheller Wolf, U. Yechiali. Fundamental Characteristics of Queues with Fluctuating Load, Technical Report CMU-CS-06-117, School of Computer Science, Carnegie Mellon University, 2006.Google ScholarDigital Library
- P. Harrison, H. Zatschler. Sojourn Time Distributions in Modulated G-Queues with Batch Processing, QEST 2004, Enschede, Netherlands, Los Alamitos, IEEE Computer Soc, pp. 90--99, 2004. Google ScholarDigital Library
- D. P. Heyman. On Ross's Conjectures about Queues with Non-Stationary Poisson Arrivals, J. Appl. Prob. 19, pp. 245--249, 1982.Google ScholarCross Ref
- C. Knessl, Y. P. Yang. An Exact Solution for an M(t)/M(t)/1 Queue with Time-Dependent Arrivals and Service, QUESTA 40, pp. 233--245, 2002. Google ScholarDigital Library
- D. M. Lucantoni, K. S. Meier-Hellstern, M. Neuts. A Single Server Queue with Server Vacations and a Class of Non-Renewal Arrival Processes Adv. App. Prob. 22, pp. 676--705, 1990.Google ScholarCross Ref
- D. M. Lucantoni, M. Neuts. Some Steady-State Distributions for the MAP/SM/1 Queue, Commun. Statist.-Stoch. Mod. 10(3), pp. 575--598, 1994.Google ScholarCross Ref
- W. A. Massey. Asymptotic Analysis of the Time Dependent M/M/1 Queue, Math. of OR 10(2), pp. 305--327, 1985.Google ScholarDigital Library
- I. Mitrani, R. Chakka. Spectral Expansion Solution for a Class of Markov Models: Application and Comparison with the Matrix-Geometric Method, Perf. Eval. 23(3), pp. 241--260, 1995. Google ScholarDigital Library
- N. Miyoshi, T. Rolski. Ross Type Conjectures on Monotonicity of Queues, Festschrift for D. Daley, P. Pollet and P. Taylor Eds. Australian & New Zealand J. of Stat. 46, pp. 121--132, 2004.Google Scholar
- M. Neuts. The Single Server Queue with Poisson Input and Semi-Markov Service Times, J. Appl. Prob. 3, pp. 202--230, 1966.Google ScholarCross Ref
- M. Neuts. The M/M/1 Queue with Randomly Varying Arrival and Service Rates, OPSEARCH 15(4), pp. 139--168, 1978.Google Scholar
- G. F. Newell. Queues with Time-Dependent Arrival Rates I -- The Transition Through Saturation, J. Appl. Prob 5, pp. 436--451, 1968.Google ScholarCross Ref
- G. F. Newell. Queues with Time-Dependent Arrival Rates II -- The Maximum Queue and the Return to Equilibrium, J. Appl. Prob 5, pp. 579--590, 1968.Google ScholarCross Ref
- G. F. Newell. Queues with Time-Dependent Arrival Rates III -- A Mild Rush Hour, J. Appl. Prob 5, pp. 591--606, 1968.Google ScholarCross Ref
- V. Ramaswami. The N/G/1 Queue and its Detailed Analysis, Adv. Appl. Prob. 12, pp. 222--261, 1980.Google ScholarCross Ref
- K. L. Rider. A Simple Approximation to the Average Queue Size in the Time-Dependent M/M/1 Queue, JACM 23(2), pp. 361--367, 1976. Google ScholarDigital Library
- T. Rolski. Queues with Non-Stationary Input Stream: Ross's Conjecture, Adv. Appl. Prob. 13, pp. 603--618, 1981.Google ScholarCross Ref
- S. Ross. Average Delay in Queues with Non-Stationary Poisson Arrivals, J. Appl. Prob. 15, pp. 602--609, 1978.Google ScholarCross Ref
- B. Sengupta. A Queue with Service Interruptions in an Alternating Random Env., OR 38(2), pp. 308--318, 1990. Google ScholarDigital Library
- B. Sengupta. The Semi-Markov Queue: Theory and Applications, Commun. Statist.-Stoch. Mod. 6(3), pp. 383--413, 1990.Google ScholarCross Ref
- T. Takine, Y. Matsumoto, T. Suda, T. Hasegawa. Mean Waiting Times in Nonpreemptive Priority Queues with Markovian Arrival and i.i.d. Service Processes, Perf. Eval. 20, pp. 131--149, 1994. Google ScholarDigital Library
- T. Takine, B. Sengupta. A Single Server Queue with Service Interruptions, QUESTA 26, pp. 285--300, 1997. Google ScholarDigital Library
- Y. Yang, C. Knessl. Asymptotic Analysis of the M/G/1 Queue with Time-Dependent Arrival Rates, QUESTA 26, pp. 23--68, 1997. Google ScholarDigital Library
- U. Yechiali, P. Naor. Queueing Problems with Heterogeneous Arrivals and Service, OR 19(3), pp. 722--734, 1971.Google ScholarDigital Library
Index Terms
Fundamental characteristics of queues with fluctuating load
Recommendations
Fundamental characteristics of queues with fluctuating load
Performance evaluation reviewSystems whose arrival or service rates fluctuate over time are very common, but are still not well understood analytically. Stationary formulas are poor predictors of systems with fluctuating load. When the arrival and service processes fluctuate in a ...
Stochastic inequalities for M/G/1 retrial queues with vacations and constant retrial policy
In this paper, we consider an M/G/1 retrial queue with vacations and we derive several stochastic comparison properties in the sense of strong stochastic ordering and convex ordering. The stochastic inequalities provide simple insensitive bounds for the ...
A Random Family of Queueing Systems with a Dynamic Priority Discipline
We consider a family {Σ u , 0 ≤ u ≤ ∞} of single server queueing systems whose customers belong to two priority classes. They arrive in two independent Poisson processes and their service times are independent with general distributions for each ...
Comments